Question

The chance of throwing 7 in \(\mathrm{a}\) wingle throw with two dice is ...

Short answer

The probability of throwing a 7 is \(\frac{1}{6}\).

Step-by-step solution

Calculate Total Possible Outcomes

When you throw two dice, each die has 6 faces. Therefore, the total number of possible outcomes is the product of the number of outcomes for each die. Thus, the total number of outcomes is \(6 \times 6 = 36\).

Identify Successful Outcomes for a Sum of 7

Next, we need to determine the combinations of two dice that result in a sum of 7. These combinations are: (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1). Hence, there are 6 successful outcomes.

Calculate the Probability of Throwing a 7

The probability of an event occurring is given by the ratio of the number of successful outcomes to the total number of possible outcomes. Therefore, the probability \(P\) of throwing a 7 is \(\frac{6}{36}\).

Simplify the Probability

Simplify \(\frac{6}{36}\) to its lowest terms. Divide the numerator and the denominator by 6, yielding \(\frac{1}{6}\).

Key concepts

Total Possible Outcomes

When rolling two six-sided dice, each die shows a number between 1 and 6. We want to figure out how many different results we can get in total from these two dice. For this, we need to calculate the total possible outcomes. Since each die has 6 sides, the total number of combinations when we roll two dice is the product of the outcomes of each die. Hence, we multiply 6 outcomes from the first die by 6 outcomes from the second die: \(6 \times 6 = 36\).
This means that there are 36 possible results when two dice are rolled. Having an understanding of total possible outcomes helps in finding the probability of specific events, like rolling a sum of 7. Remember, every possible combination of dice values is equally likely, and thus, calculating total possibilities forms the basis of probability with dice.

Successful Outcomes

To find the probability of throwing a sum of 7 with two dice, we must identify which combinations of dice values lead to this total. A successful outcome here means having two numbers on the dice that add up to 7.
Let's list these combinations:

• (1, 6)
• (2, 5)
• (3, 4)
• (4, 3)
• (5, 2)
• (6, 1)

As there are six combination pairs that result in a sum of 7, we have 6 successful outcomes. Recognizing these successful outcomes is crucial because they are necessary to determine the probability of rolling a sum of 7 by comparing them to the total possible outcomes. The key here is to systematically list all combinations and ensure none are overlooked.

Simplifying Fractions

Once we've determined both the total possible outcomes and the successful outcomes, calculating probability becomes straightforward. For the probability, we take the number of successful outcomes (6) and divide it by the total possible outcomes (36), giving us a fraction: \(\frac{6}{36}\).

Simplification of this fraction involves reducing it to its simplest form. To simplify any fraction, find the greatest common divisor (GCD) of the numerator and the denominator. Here, the GCD of 6 and 36 is 6. When we divide both the numerator and the denominator by 6, we arrive at: \(\frac{1}{6}\).
This simplified fraction still represents the same probability, but in an easier-to-understand form. Simplifying fractions is a useful practice in probability to make results more understandable, especially for more intuitive interpretation in real-world applications.